Polymeric fractals and the unique treatment of polymers

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Polymeric fractals and the unique treatment of polymers

T.A. Vilgis

To cite this version:

T.A. Vilgis. Polymeric fractals and the unique treatment of polymers. Journal de Physique, 1988, 49

(9), pp.1481-1483. �10.1051/jphys:019880049090148100�. �jpa-00210828�

1481

Short communication

Polymeric fractals and the unique ^{treatment of} polymers

T.A. Vilgis

Max-Planck-Institut für Polymerforschung, ^PO-Box 3148, ^D-6500 Mainz, ^F.R.G.

(Re§u ^le 24 ^mai 1988, accepti ^le 4. juillet 1988)

Résumé.- Il est suggéré que les r-fractals permettent ^une description unifiée des polymères ^connectés

arbitrairement. r est l’exposant ^du rayon de giration ^maximum possible, R ~ ^{Nr. Nous} conjecturons

que 1/r est ^identique à la dimension spectrale ^{du fractal} polymère.

Abstract.2014 It has been suggested that r-fractals allow a uniform description ^of arbitrarily ^connected polymers. ^{r is the} exponent of the maximal possible ^{radius of} gyration, i.e. R ~ Nr. We conjecture

that 1/r is identical with the spectral dimension of the polymeric ^fractal.

Tome 49 N° 9 SEPTEMBRE 1988

LE JOURNAL DE PHYSIQUE

J. Phys. ^{France 49} (1988) ^{1481-1483 SEPTEMBRE 1988,}

Classification

Physics ^Abstracts

61.40 - 05.90 - 82.70

Introduction.

In a recent paper [1] ^{it has been suggested that}

socalled r-fractals are a model for the descrip-

tion of _any polymer, ^i.e. linear, ^{branched or} any

arbitrary ^connected polymeric object. ^{Here r is}

defined by ^the scaling ^{of the} maximum radius of gyration ^{with the total mass in} the fractal

N, i.e. R N Nr. ^r is assumed to be a superu- niversal exponent, ^i.e. independent ^{of the Eu-}

clidian _space dimension d. Moreover this _paper contains an interesting discussion of the general problem ^{of the use of} Flory-de ^{Gennes theories}

by ^a mean-field estimation of fractal dimensions of such polymers. Unfortunately ^this paper has

some shortcomings, ^for example ^it suggests ^that

the spectral dimension of the generally ^linked polymer depends ^{on the} space dimension via the

two-body repulsion ^{and on} the value of r itself.

Since in "ordinary" fractals, ^i.e. non-polymeric fractals, ^the spectral ^{dimension is an intrinsic}

parameter ^and depends only ^on the connecti-

vity, ^this point ^{is left} unsatisfactory. ^{In this short}

communication we want to suggest ^{a consistent} picture which does not lead to such a contra- diction. We suggest ^that the inverse of r is the

spectral dimension itself.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049090148100

1482

Polymeric fractals constructed from lat- tice fractals and Flory theory [2].

The Gaussian fractal dimension of the ideal Gaussian polymeric ^{fractal can be} defined, by replacing ^the (rigid) ^{bonds of an} arbitrarily

constructed lattice fractal by highly ^flexible

Gaussian polymer ^chains [2]. ^An important point is that the connectivity ^{of the} original

lattice fractal is preserved by ^this procedure.

The spectral dimension is an intrinsic parameter which is determined by ^the connectivity only [3]

is then preserved ^for ever, i.e. no thermodynamic

process, like swelling ^or diluting, ^has any effect in the connectivity ^{or on the} spectral ^dimension.

Using ^the general Alexander-Orbach relation [3]

between fractal-, spectral-, and walk dimension

(df, dg, dW respectively) ^the resulting ^Hausdorff-

or fractal dimension df ^{of the} (Gaussian) ^poly-

meric fractal (defined ^by the size - mass scaling

Rgf I"V M) ^{is given by} [2,4]

The well known case of linear chains is recovered

immediately by choosing ds ^{= 1.} The mean-field value of the spectral dimension of percolation

clusters ^or lattice animals is ds ⁼ 4/3 [3], ^and

the fractal dimension is then predicted correctly

to 4, corresponding ^{to the} Stockmayer - Flory

limit. Note that on the level of standard Flory arguments, ^{which we are} going ^to apply now,

only ^the mean-field values of d. ^are required.

The important point of the existence of the Gaussian fractal dimension is that excluded volume interactions can be taken into account

by ^usual Flory arguments [5]. ^{Here we use the}

classical approach ^{for reasons of} simplicity only

and do not follow the more general ^discussion

as presented in reference [1]. ^{The Flory free}

energy of the polymeric fractal is given by ^the entropic ^elastic part, Fej - (RI Ro)2, ^{and by}

a two-body repulsion energy U N vN2 I Rd.

R is the size of the object, ^v the excluded volume parameter, ^{and N the total mass of the} abject. Ro corresponds ^to the Gaussian size of the fractal under ideal conditions, i.e. in absence of excluded volume forces. Minimization of the total free _energy, Fel ⁺ U, predicts the swollen fractal dimension in dilute solution [2]

The _upper critical dimension (UCD) ^{can be}

found easily by estimating ^{the size} dependence

of the two-body potential, ^{i.e. a} Ginzburg ^crite-

rion. Then the UCD is given by [2] ^{due =} 4ds, 2-d,

which predicts ^with d. ⁼ 4/3 ^{the UCD of bran-}

ched polymers due ^{= 8 and with} ds ⁼¹ du, ^{= 4}

for linear chains correctly.

Moreover we mention that, by ^this simple Flory arguments, ^{we can} predict mean-field frac- tal dimensions in dense systems, ^{such as melts} of polymeric ^{fractals or} mixtures of polymeric

fractals of different fractal dimension, ^{if we take} screening of excluded volume forces into account

[6,9].

Connection to r-fractals.

These equations ^{have to be} compared ^{to the}

results obtained in reference [1]. ^{First we note}

that the spectral dimension in equation (1) ^and

equation (2) is fixed forever and cannot depend

on the dimension of the embedding space, ^as it must be. As a further remark we mention that it is a priori not neccessary to construct ^a

polymeric fractal from a lattice fractal. It was

just ^{a tool to} calculate the Gaussian fractal dimension df ^{of the} corresponding ^{fractal. Note}

further that equation (1) ^{is a general expression}

for this Gaussian dimension for _any connected

polymer ^with spectral ^dimension ds .

It is important ^to realize that most of the

physical results of the approaches in reference

[1] and reference [2] ^are consistent if we require

r = lld.. ^{We are going to} demonstrate now that this conjecture ^is physically ^sensible.

Fractal dimension and UCD

The swollen fractal dimension in the limit of the

simple Flory-de ^Gennes theory (k ^{= 2 in Ref.}

[1]) for the r-fractal was calculated to

It is immediately ^{seen that} (3) ^and (2) ^are

identical if we assume 1/r ⁼ dg . ^{With this}

choice the UCD, i.e. that dimension where the

size dependence ^{of the} interaction potential ^{is no}

longer important ^{is identical as} well, ^{and both}

approaches ^are consistent.

1483 Lower critical dirriension (LCD) ^{and r}

Consider now the lower critical dimension

(LCD). ^{This can be defined} generally by putting Df ⁼ d, i.e. it is that dimension where the effect of the excluded volume effect is most severe. As

a remark beside, ^{note that the same condition} produces ^the saturation effect in some cases, i.e.

defines a transition from fractal to non-fractal behaviour [7,8]. ^{This is important in cases where}

mixtures of different polymeric ^{fractals or melts}

of polymeric ^{fractals are} considered [8]. ^{In the}

case of good ^solvent Df = ^d gives ^{the LCD.}

Equation (3) predicts ^{for the LCD}

whereas equation (2) predicts

If we insert this LCD in the swollen fractal di- mension we find the minimum fractal dimension,

or equivalently ^{the maximum radius of} gyration.

It is given by

i.e. R N N for linear chains and R - N 3/4

for branched polymers. ^{This was assumed in}

reference [1] ^{to be a} super-universal exponent.

Here we offer that r is determined by ^the spec- tral dimension and has to be interpreted ^{in the}

sense that the spectral dimension is given by ^the connectivity only. Moreover, ^{once it is fixed} by

the connectivity ^{it does not} depend ^{on the diffe-}

rent swelling ^{behaviour in} different Euclidian di-

mensions, ^{due to the} repulsive energy. We want to stress that this discussion does not correspond

to the question ^{of the} validity of the Alexander- Orbach conjecture, ^since (as already ^mentioned

above) ^{we have to use} only the mean-field values in the Flory-de ^Gennes type arguments.

Ideal Gaussian dimensions

The same result is consistent with the prediction

of reference [1]

where _vo is the correlation length exponent ^of the ideal Gaussian fractal, ^i.e. 1/2 for linear chains or 1/4 ^{for branched} polymers ^{in the} Zimm-Stockmayer limit. If we put 1/vo = ^df,

i.e. the Gaussian fractal dimension equation (6)

is exactly ^{the same as} equation (1) ^{if one uses the}

Einstein relation for ideal polymeric ^fractals [2]

dw = df + 2, ^{i.e. the} resistivity scaling exponent for Gaussian fractals is 2 (see ^Ref. [2] ^{for a clear} discussion).

Discussion.

All these arguments give ^a consistent picture

with reference [1] ^{but do not lead to a strange}

behaviour of the intrinsic exponents, ^{such as the} spectral dimension of the fractal dimension of the chemical distance. Surely ^{we have} presented only ^crude conjectures within this short commu-

nication, ^and they ^{remain to be} proven. Never- theless we suggest ^{that the} exponent r-1 ^intro- duced in reference [1] ^{is the} spectral ^dimension

of the polymeric fractal itself. We find no contra- diction to this conjecture and is consistent with

previous ^results [2,6].

Acknowledgement.

The author is grateful ^{to C.} Schrauwen for his critical remarks and help ^{on an} earlier version of the manuscript.

References

[1] LHUILLIER, D., ^J. Phys. ^{France 49} (1988)

705.

[2] CATES, M.E., ^J. Phys. ^{France 46} (1985)

1059.

[3] ALEXANDER, ^{S. and} ORBACH, R., ^J. Phys.

France 43 (1982) ^L2014625.

[4] Note that the Einstein relation for polyme-

ric fractals is dw ⁼ df + 2 [2].

[5] ^DE GENNES, P.-G., Scaling Concepts ⁱⁿ

Polymer Physics (Cornell University Press, Ithaca) ^1979.

[6] VILGIS, T.A., Phys. Rev. A. 36 (1987) ^1506.

[7] VILGIS, T.A., Makromol. Chem. Rapid

Comm. (1988) ^{in press.}

[8] VILGIS, T.A., (1988) ^preprint.

[9] VILGIS, T.A., ⁱⁿ Comprehensive Polymer

Science 5, ^{Eds. G.C. Eastmond et al.} (Per-

gamon Press) ^{to appear 1988.}